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प्रश्न
Using Ampere's Law, derive an expression for the magnetic induction inside an ideal solenoid carrying a steady current.
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उत्तर
- Consider an ideal solenoid as shown in the figure below.
Ampere’s law applied to a part of a long ideal solenoid - The dots (.) show that the current is coming out of the plane of the paper and the crosses (×) show that the current is going into the plane of the paper, both in the coil of square cross-section wire.
- For the application of Ampere's law, an Amperian loop is drawn as shown in the figure and box.
- Using Ampere’s law,
`ointvec"B".vec"d""l" = mu_0"I"`
Over the rectangular loop abcd, the above integral takes the form
`int_"a"^"b" vec"B".vec"d""l" + int_"b"^"c" vec"B".vec"d""l" + int_"c"^"d" vec"B".vec"d""l" + int_"d"^"a" vec"B".vec"d""l" = mu_0"I"`
where, I is the net current encircled by the loop.
∴ BL + 0 + 0 + 0 = `mu_0"I"` ….(1)
Here, the second and fourth integrals are zero because `vec"B"` and `vec"d""l"` are perpendicular to each other. The third integral is zero because outside the solenoid, B = 0. - If the number of turns is n per unit length of the solenoid and the current flowing through the wire is i, then the net current coming out of the plane of the paper is
I = nLi
∴ Using equation (1)
BL = `mu_0"nLi"`
∴ B = `mu_0"ni"` ….(2)
Equation (2) is the required expression.
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संबंधित प्रश्न
A solenoid of length π m and 5 cm in diameter has winding of 1000 turns and carries a current of 5 A. Calculate the magnetic field at its center along the axis.
A toroid of a central radius of 10 cm has windings of 1000 turns. For a magnetic field of 5 × 10-2 T along its central axis, what current is required to be passed through its windings?
What is Solenoid?
What is Toroid?
A toroid of 4000 turns has an outer radius of 26 cm and an inner radius of 25 cm. If the current in the wire is 10 A. Calculate the magnetic field of the toroid.
A solenoid of length π m and 5 cm in diameter has a winding of 1000 turns and carries a current of 5 A. Calculate the magnetic field at its centre along the radius.
The length of solenoid is I whose windings are made of material of density D and resistivity p. The winding resistance is R. The inductance of solenoid is
[m = mass of winding wire, µ0 = permeability of free space]
A winding wire which is used to frame a solenoid can bear a maximum of 20 A current. If the length of the solenoid is 80 cm and its cross-sectional radius is 3 cm, then the required length of winding wire is ______ (B = 0.2 T)
The ratio of magnetic field and magnetic moment at the centre of a current carrying circular loop is x. When both the current and radius is doubled then the ratio will be ______.
The magnetic induction along the axis of a toroidal solenoid is independent of ______.
The space within a current carrying toroid is filled with a m metal of susceptibility 16.5 x 10-6. The percentage increase in the magnetic field B is ____________.
A toroid is a long coil of wire wound over a circular core. If 'r' and 'R' are the radii of the coil and toroid respectively, the coefficient of self-induction of the toroid is (The magnetic field in it is uniform and R > > r) ____________.
(N = number of turns of the coil and µ0 = permeability of free space)
The magnetic flux near the axis and inside the air core solenoid of length 60 cm carrying current 'I' is 1.57 × 10-6 Wb. Its magnetic moment will be ______.
(cross-sectional area is very small as compared to length of solenoid, µ0 = 4π × 10-7 SI unit)
Magnetic field at the centre of a circular loop of area 'A' is 'B'. The magnetic moment of the loop will be (µ0 = permeability of free space) ____________.
A long solenoid carrying current 'I1' produces magnetic field 'B1' along its axis. If the current is reduced to 25% and number of turns per cm are increased four times, then new magnetic field 'B2' is ____________.
A winding wire is used to prepare a solenoid that can bear a maximum current of 10A. If the length of a solenoid is 80 cm and its cross-sectional radius is 3 cm, the required length of winding wire is ____________.
(magnetic field B = 0.2 T, µ0 = 4 x 10-7 SI units)
A charged particle carrying a charge 'q' and moving with velocity V, enters into a solenoid carrying a current T, along its axis. If 'B' is the magnetic induction along the axis of a solenoid, then the force 'F' acting on the charged particle will be ____________.
In a current-carrying long solenoid, the field produced does not depend upon ______
A straight solenoid has 50 turns per cm in primary and 200 turns per cm in the secondary. The area of cross-section of the solenoid is 4 cm2. The mutual inductance is ______.
Obtain an expression for magnetic induction of a toroid of ‘N’ turns about an axis passing through its centre and perpendicular to its plane.
For a solenoid and a toroid, the number of turns per unit length is n and the respective interior volume is V. The self inductance is proportional to n2 and V for ______.
A current of 10 A passes through a coil having 5 turns and produces magnetic field at the centre of the coil having magnitude 0.5 x 10-4 T. Calculate diameter of the coil.
(µ0 = 4π x 10-7 Wb/Am)
